1
$\begingroup$

Taylor series can create harmonics of the frequencies in an input signal. I'm wondering if it is likewise possible to use Volterra series to create sub-harmonics of the frequencies in that signal.

Some degree of finesse is required in posing the question:

Taylor series don't only create harmonics; there's intermodulation distortion as well. Likewise, it's ok with me if a suitable Volterra series doesn't only create sub-harmonics, but also creates something similar to intermodulation distortion, or even some degree of harmonic distortion.

However, I would like to exclude trivial cases where the input signal gets so wrecked that it ends up looking like white noise, which would obviously create sub-harmonics (along with every other frequency you can imagine!). I'm not sure how to formalize this requirement precisely, other than to say that I hope it's clear what I'm driving at.

Here's one way to formalize the behaviour I want:

  1. Given an input $\cos(\omega t)$, yields an output containing $\cos(\frac{\omega}{n} t)$ for some fixed $n$, ideally a natural number
  2. Given a general input, a sub-harmonic is generated corresponding to every frequency in the input
  3. Other "intermodulation distortion"-type artifacts are allowed, "within reason"

The answer can be expressed for either discrete or continuous signals.

$\endgroup$
  • $\begingroup$ interesting question. i don't know for sure, but i suspect that you cannot generate sub-harmonics without a time-varying component in the algorithm. Volterra is non-linear, but is also time-invariant. $\endgroup$ – robert bristow-johnson Nov 14 '15 at 14:25
2
$\begingroup$

Check out this paper. I would have made a comment but not high enough rep.

http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=1211087&url=http%3A%2F%2Fieeexplore.ieee.org%2Fiel5%2F81%2F27258%2F01211087

Looks like you need multiple in to get subharmonics in Volterra series

The abstract states "Subharmonic generation is a complex nonlinear phenomenon which can arise from nonlinear oscillations, bifurcation and chaos. It is well known that single-input-single-output Volterra series cannot currently be applied to model systems which exhibit subharmonics. A new modeling alternative is introduced in this paper which overcomes these restrictions by using local multiple input single output Volterra models. The generalized frequency-response functions can then be applied to interpret systems with subharmonics in the frequency domain."

$\endgroup$
  • $\begingroup$ Thanks, although I'm a bit confused. When we talk about a MISO system, which input are we generating subharmonics of? Both? I can't see the paper, so I'm a bit confused here. $\endgroup$ – Mike Battaglia Nov 19 '15 at 9:26
  • $\begingroup$ full disclosure - I have never done MISO Volterra as I can limit my analysis that I do to SISO but I would imagine that you could look at analygous to a subharmonic mixer if you have ever came across one. In this case, a low frequency tone is mixed with a high frequency one, say something like LO = 10 GHz and RF = 100.1 GHz, the mixer product, IF would be at multiple subharmonics but if you wanted the 10 harmonic of the LO, IF would be 100 MHz ( this is done a lot with millimeter wave systems) - but really, I would read the paper I linked here. $\endgroup$ – johnnymopo Nov 20 '15 at 3:05

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.